Subformula property in many-valued modal logics

Abstract

Fitting, in [1] and [2], investigated two families of many-valued modal logics. The first, which is somewhat familiar in the literature, is that of the logics characterized using a many-valued version of the Kripke model (binary modal modelin his terminology) with a two-valued accessibility relation. On the other hand, those logics which are characterized using another many-valued version of the Kripke model (implicational modal model), with a many-valued accessibility relation, form the second family. Although he gave a sequent calculus for each of these logics, it is far from having the cut-elimination property (CEP) or the subformula property. So we will give a substitute for his system enjoying the subformula property, though it is not of ordinary sequent calculus but of the many-valued version of sequent calculus initiated by Takahashi [7] and Rousseau [3].The author, unaware of the deduction systems with CEP, had given in [8] and [9], after Rousseau [4], the deduction systems for the intuitionistic many-valued logics which enjoy CEP only for a certain restricted class of proofs. Then in [10], he gave for three-valued modal logics the ones with CEP, but these systems have a rule of inference which is unnecessary if the Cut rule is present. Why are we particular about CEP? The author's answer is that a cut-free proof is easy to examine since it is composed solely of subformulas of the formulas which form its conclusion. In this direction, the author has given, for modal logics with the Brouwerian axiom [11], the ones without CEP which nevertheless enjoy the subformula property. This paper is a sequel to the study in [11].